Novel routes to chaos through torus breakdown in non-invertible maps

Zhanybai Zhusubaliyev, Erik Mosekilde

Research output: Contribution to journalJournal articleResearchpeer-review


The paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non-invertible maps. By means of detailed, numerically calculated phase portraits we first describe how three- and five-layered tori arise through period-doubling and/or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. We then describe several different mechanisms for the destruction of five-layered tori in a system of two linearly coupled logistic maps. One of these scenarios involves the destruction of the two intermediate layers of the five-layered torus through the transformation of two unstable node cycles into unstable focus cycles, followed by a saddle-node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set. Other scenarios involve different combinations of local and global bifurcations, including bifurcations that lead to various forms of homoclinic and heteroclinic tangles. We finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two-dimensional non-invertible maps that have previously been used to study the properties of invariant sets.
Original languageEnglish
JournalPhysica D : Non-linear Phenomena
Issue number5
Pages (from-to)589-602
Publication statusPublished - 2009


  • Torus breakdown
  • Non-invertible maps
  • Multilayered torus
  • Coupled logistic maps


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